Apparatus for approximately squaring circles



Jan. 11, 1949.

I I I I I I b m l I I I l I I I I I I I APPARATUS FOR APPROXIMATELY SQUARING CIRCLES Filed July 8, 1348 Sheets-Sheet 1 w/ I I I /2 I I I I I I I I I I I Q- aa; \D A D] N I I s i \I Ch I -I LE1 INVHVTOR.

ATTOF? NEYS Jan. 11,1949. w THOMSON 2,458,749

APPARATUS FOR APPROXIMATELY SQUARING CIRCLES Filed July 8, 1948 3 Sheets-Sheet 2 a T SIDE SIDE AREA AREA E\ 0 0 D] m Q\\ L Q k m N N -zg L: k I l l R: w Q *T S u.

\ 0, r" L I I a m Q I a b a INVENTOR.

WILLIAM B.THOM$ON W M fl ATTORNEYS Jan. 11, 1949. w, THOMSON 2,458,749

APPARATUS FOR APPROXIMATELY squxnme CIRCLES Filed July ai 1948 3 Sheets-Sheet. 3

INVENTOR.

WILLIAM B. THOMSON BY 2 g ATTORNEYS Patented Jan. 11 1949 UNITED STATES OFFICE- APPARATUS FOR APPROXIMATELY SQUARI'NG CIRCLES William Balvaird Thomson; -Biiston; Mass.

ApplicationrJulys, 1948, SeriaLNo; -37',594-

7 Claims. 1,

The present invention-relates to approximative circle-squaring, andmore particularly to apparatus for determining the length of the side. of the squarehaving substantially the same area as any given circle. Such a square will be referred to hereinafter as the equivalent square.

An: object of the invention is to provide a new and improved instrument for approximately squaring the circle.

Other and further objects will=-be explained hereinafter and will be more particularly pointed out in the appended claims.

The invention will now be described more fully in connection with the accompanying drawings, in which Fig. 1 is a perspective, partly broken away, of a preferred instrument embodying the present invention, shown applied to a circular cylinder for the purpose of determining the lengthof the side of the equivalent square having approximately the area. of the circular base of the cylinder and showing also mathematical-symbolsas amaid to an understandingof the principles'underlying the invention LFig. 2 is'a diagramillustrative ofthe principles and explanatory ofthe mathematical symbols; Fig. 3 is a plan ofthe instrument shown inFig. 1, also partly broken away, and:

showing also the mathematical symbols; Fig. 4 is anend view of the instrument shown in Fig.4, looking from the left; Fig. 5 is a section: of the slidable indicator member of the preferred instrument upon a larger scale; taken upon the line 5 5 of Fig. 3, lookingin the direction of the arrows; and Fig, 6 is an underside perspective of this slidable indicator member, partlybroken-away, and upon alarger scale.

The preferred instrument of the present inventionmay comprise a relatively large base member I"! and a relatively smallindicator mem-- ber 23; These members I! and 23 are both shown as rectangular plates, disposed in:parallel planes at right angles to each other, and relatively slidable in these parallel planes along adirection parallel to the longer side edges 33 and-35 of the rectangle of the base member H. I For definiteness,.the base member I1 will be referred to as a stationary member and the indicator member 23 as a slidablemember.

The stationary member I! and the slidable indicator member 23 are provided with right-angunature of which will be explained: more fully here'- in'aft'er. The straight-line scale I', with itscali-- bratio'n's 'fi', theinolinedlihe 3, the column headed I'nch. and the columns described more fully h'e'reinafter'm'ay be disposed in a rectangle l B; in

th'e'plarre of t'l'ie'upper'orfrontface l5of the base memb'erlll As shown at 6 and H; respectively; the scales I and 12" may be" calibrated linearly in the same units'ffor exampl'e', inches; The scales 1' and I2" may be provided with suitable numerals to indi-- catetheva'lues corresponding to the respective calibrations "6 and H. Inth'e"'co1umn headed" projectingiend' portion2 l'is shown provided with a plane'iace 22, parall'el tothe shorter sid'es'of therectangle'of"the stationary'base member fl' a'ndperpendic'ular to the" planes oi *the rectangles of the base member I! and the indicator member 231- Along'dt's left-hand longer side; the indicator'membe'rfiu's provided with a similarly pro=- je'cting" portion 26 havinga; plane f ace Mparallel anddisposedoppositeto theplane face 22'. The pla'nes'of thefaces 22 and 26 aretherefore' perpendiculartothe direction of' the slidable movement'ofthe'indicator member 23;

Thepoint ofintersectionpf' the straight-line scale I and the inclirie'dntraightline 3 is prefer= ably disposed'ihthe'plane of the face 22; Asthe straight liire-scale' [may-be calibrated to read zero at the point 1" ohm intersection with theinclined straight-lineil, in the plane of the face'22'j this point T may be termed the'originofthe straightline-sca-lel. The straight-line" scale IZis preferably disposed i'ntheplaneof the faceZl. As-the stral'ght line scale l 2may similarly'becalibrated to read-zero atthe-point 9 of its intersection with the straight-linescale l', point- 9' may similarly -be=--termed*th'e origin of the straight-line scale I 2. The readingindicatedmy the point [3f on'the straight-line scaietfrepresents the distance between the origin 9 of: thescale l2, which is the foot of the perpendicular to the straight-line scale I throu'ghthe' point I'3', and'thep'oint l3'of intersection of the perpendicular with the inclined straight line 3; I This-reading will be represented by the Quantity 23:;

At diif'erent positions of slidable adjustmentof the indicator member 23',- .the point 9 of the straight-line scale l2 will indicate in the column headed Inch different readings of the calibrated values of thestraight-line scale I. Since the parallel plane faces 22 and 21 corresponding to different positions of adjustment of the plane face 21 toward and from the plane face 22.

Assuming, as an illustration, that it is desired to determine the length 2s of the side of the equivalent square having substantially the same area as the cross-sectional area of a circular cylinder 5 of radius a, the face 22 of the right-hand projecting end portion 2| of the stationary member ll may first be caused to engage a point of the circumference of the cylinder 5, and the indie cator member 23 may then be slidably adjusted until a diametrically opposite point of the circumference becomes engaged by the parallel face 21 of its projecting portion 26, as shown in Fig. 1. The reading in the column headed Inch of the calibrated straight-line scale I indicated by the point 9 will therefore measure the value of the diameter 2a of the circular base of the cylinder 5. From this reading 2a is subtracted the reading 211:, indicated by the point [3 on the scale [2, representing the distance from the origin 9 of the scale l2 to the point l3 of intersection of the scale I2 with the inclined straight line 3. The value Za-Za: obtained by this subtraction, assuming the suitable value for the angle o of inclination between the straight-line scale I and the straight line 3, as will presently be demonstrated, is the length 28 of the side of the desired equivalent square.

If the radius a of the circular base of the cylinder 5 is 4 inches, for example, the reading of the scale I indicated by the origin 9 of the scale l2 willbe 211 or 8 inches. Assuming the proper value of the angle the reading 2a: of the scale [2 indicated by the point l3 will then be found to be 0.91 inch. The difference between 2a-2w is therefore 7.09 inches. This is the length 2s of the desired equivalent square.

In proof of the fact that the length 23 of the side of the equivalent square may be obtained in this manner when the straight-line scale I and the straight line 3 are inclined to each other at an angle o of suitable value, reference may now be made to Fig. 2, representing, concentrically arranged, a circle 2 of any desired radius a, and the equivalent square 4 having approximately the same area as that of the circle 2, the half-side of which is represented at s. The distance 0:, as shown in Fig. 2, represents the difference between the radius a of the circle 2 and the half-side s of the square 4:

These equations furnish data for determining wherefore the angle of inclination of the straight line 3 to the straight-line scale From an inspection of Fig. 1 or Fig. 2,

According to the present invention, therefore, the length 28 of the side of the equivalent square 4 having the same area as that of a circle 2 of radius a may be determined. All that is necessary in order to solve this problem, after adjusting the indicator member 23 along the base member I! until the circle is tangent to the parallel faces 22 and 21, is to measure, in the column headed Inch, along the straight-line scale I, from its origin 1 to the origin 9 of the straightline scale l2, the calibrated distance 2a, representing the diameter of the circle 2, then to measure the calibrated distance 21: from the origin 9 of the straight-line scale I2 to its point l3 of intersection with the inclined line 3, and then to subtract the measurement 21' from the measurement 2a.

The preferred instrument for solving this problem will now be described more fully.

It has been stated that the parallel faces '22 and 2'! are respectively provided upon one of the projecting end portions 2| of the base member I! and the projecting portion 26 of the indicator member 23. For purposes of assembly, it is desirable that at least one of these projections be detachable. According to the illustrated embodiment of the invention, the end portions 2| are shown integral with the base member I1 and the projecting portion 26 is shown separate from the indicator member 23. The rectangular indicator member 23 is shown interposed between the projecting portion 26 therebelow and a cover member 25 thereabove. The projecting portion '26 andthe cover member 25 are shown of the same rectangular configuration as that of the indicator member 23. The cover member 25, the indicator member 23 and the projecting portion 26 may be secured-together in any desired way, as by means of screws Ill. The cover member 25 is shown in the form of a frame provided with a rectangular opening through which to view the scale 12 of the indicator member 23.

- The-base member I! is slidably received in a suitably shaped recess of the projecting portion 23. In transverse section, this base-memberreceiving recess is shown U-shaped, the bottom wall of the U being indicated at '24 and the side walls of the U at 29 and 3|. The lower or rear face H! of the base member I! contacts slidably with the bottom wall 24 of the U-shaped recess. The side edge 33 of the base member I! similarly contacts slidably with the side wall 29 of the U- shaped recess. The side edge 35 of the base member l1, however, is shown separated slightly from the side wall 3| of the U-shaped recess in order to provide a space for receiving a leaf-spring clip 4|. The leaf-spring clip 4| is intermediately fastened, as by means of a screw l4, to the side wall 3| .of the U.-shaped recess, and the ends of the leaf-spring clip 4| bear against the side edge 35 at thebase member ll. The leaf-spring clip ll thus serves to hold the indicator member '23 snugly upon the base member 11 in any position of its slidable adjustment.

Accidentaldetachment of the indicator 'member 23 from the base member l1 atthe limits of its slidable movement is prevented by the projecting portion 26 of theindicator member 23 engaging the projecting end portions 2i of the base member ll.

It has been stated that the calibrations 8 of the straight-line scale I and the inclined line 3 are provided in the plane of the upper or front face 15 of the base member'l. In order to minimize errors of reading due to parallax, thecallbrations ll of the straight-line scale l2 may be provided upon the lower or rear face 8 of the indicator member 23, adjacent and opposite to the front face I 5 of the besemember ll. In order not to hide from view the'calibrations II of the straight-line scale [2, the indicator member 23 may be constituted of a sheet of transparent material, suchas glass or plastic. The indicator member 23 could, of course, be opaque and the base-member ll could-be'made transparent. In either event, this construction renders it possible to position the calibrations I I of the scale l2 upon the lower face 8 of the indicator 23 as close as may be desired to-the calibrations 6 of the scale I and the inclined line 3 upon the upper surface l5 of the base member H. In order to prevent the adjacently disposed faces 8 and I5 rubbing against each otherduring the slidable movement of indicator member 23 on the base member l1, very thin shims 31' and 39 may be interposed along the margins of the small side of the rectangle of the indicator member 23, between the upper face of the projecting portion 26 and the lower face 8 of the indicator member 23. These shims 31 and 39 are shown wide enough so that their inner edges aline with the long sides of the rectangle H5 in the upper face ii of the base member IT. The shims 31 and 39 are shown held in place by the same screws 10 that hold the indicator member 23 between the cover 25 and the projecting portion 26.

For purposes of check, it is convenient to provide tables upon the front face I5 of the member 11 in the columns headed Side and Area. These columns are shown disposed along with the column headed Inch, the straight-line scale I and the inclined line 3 in the rectangle IS. The two columns headed Side, for example, may contain tabulations of the values of the lengths 2.: of the sides of theequivalent squares corresponding to the readings on the scale I of the circle diameters 2a, tabulated in the column headed Inch. Corresponding to the circle-diameter reading 1 inch of the calibrated scale I in the table headed Inch, as an illustration, the value 0.8862, indicating the length of the side of the equivalent square, may be tabulated at 45. The corresponding area 078540 of the circle or its equivalent square may be tabulated in the corresponding column headed Area at 41. In those cases where the reading of the instrument corresponds to a circle diameter between the calibratic-ns of the scale I, in the column headed Inch, it is necessary, of course, to calculate the value of this area. The tabulations, however, afford a rough check upon the calculations.

Values of the circle diameters 2a and the lengths 25 of the sides of the equivalent squares and of the corresponding areas may be tabulated on-the instrument. Such values may commence, for example, asfollows:

Diameter Area 1a or (2) Side 2.

Inches Inches Square inches M a 0. 0554 00307 O. 1108 01227 its 0. 1662 02761. $41 0. 2216 04009 545 0. 2769 07070 0. 3323 11045 Ms 0.3877 15033 3 0. 4431 19635 As 0. 4985 24850 541 0.5539 30680 "As 0. 6093 .37122 0. 6647 44179 %0 0. 7201 51849 Z6 0. 7754 60132 It,is not essential, of course, that the calibrations corresponding to the circle diameters be provided upon the straight-line scale I. They could equally well be provided, for example, upon any other straight line through the origin 1 and, in particular, the straight line 3. The distance between the origin 1 and the point of intersection of such other straight line with the straight-line scale I2 is as good a measure of the diameter 2a of'the circle 2 as the distance between the origin 1 and the point "9. If such other straight line be calibrated linearly in units the same as the units of calibration of the straight-line scales 1 and I2, but multiplied by the secant of the angle of inclination of such other straight line to the straight-line scale .I, the values of the calibrations in the column headed Inch will apply to the calibrated scale of such other straight line equally well as to the straight-line scale I.

If the straight-line 3 were graduated in this manner,'for example, it would be possible to have the point 13 indicate, not only the value 2:: on the scale 12, but also the value of the diameter 2aon'the inclined straight line 3. The graduations of the inclined straight line 3 would be the same asithe graduations of the straight-line scale I, multiplled'by sec 4:, but the graduations could have the same numbers associated with them as are associated with the graduations of the straight-line scale I, as tabulated in the column headed Inchf Further modifications, too, will occur to "persons skilled inthe art, and all such are considered to fall withinthe spirit and scope of the present invention, as deflned'in the appended claims,

What is claimed is:

1. An instrument for approximately squaring a circle having, in combination, a base member having two straight lines inclined to each other at an angle determined by the equation where 11' is the ratio of the circumference to the diameter of a circle, one of the lines being linearly calibrated in predetermined units, and an indicator member movable with respect to the base member substantially parallel to one of the lines and calibrated in the same units.

2. An instrument for approximately squaring a circle having, in combination, a base member having an origin and a straight-line scale through the origin calibrated to correspond to measures of the values of the diameters of circles, and an indicator movable with respect to the base member substantially parallel to a further line through the origin and provided with a calibrated scale.

the base member having also a line through the origin inclined to the further line at an angle 4: determined by the equation.

where 1r is the ratio of the circumference to the diameter of a circle.

3. An instrument for approximately squaring a circle having, in combination, a base member having a face provided with two straight lines inclined to each other at an angle determined by the equation where 1r is the ratio of the circumference to the diameter of a circle, one of the lines being linearly calibrated in predetermined units, and an indicator member movable with respect to the base member substantially parallel to the said one line and having a face provided with a straight line at right angles to the said one line calibrated in the same units, the face of the base member being disposed adjacent and opposite to the face of the indicator member, and one of the members being transparent.

4. An instrument for approximately squaring a circle having, in combination, a base member having two straight lines inclined to each other at an angle of substantially six degrees, twenty-nine minutes, twenty-six seconds, one of the lines being linearly calibrated in predetermined units, and an indicator member movable with respect to the base member substantially parallel to the circle having, in combination, a base member having two straight lines inclined to each other at an angle of substantially six degrees, twenty-nine minutes, twenty-six seconds, one of the lines being linearly calibrated in predetermined units, and an indicator member movable with respect to the base member substantially parallel to the said one line and provided with a straight line at right angles to the said one line calibrated in the same units, the base member being provided with a table of values of the lengths of the sides of squares equivalent to circles having diameters corresponding to the calibrations of the said one line, and the said values being tabulated adjacent to the corresponding calibrations.

6. An instrument for approximately squaring a circle having,in combination, a base member haw ing two straight lines inclined to each other at an angleof substantially six degrees, twenty-nine minutes, twenty-six seconds, one of the lines being linearly calibrated in predetermined units, and an indicator member movable with respect to the base member substantially parallel to the said one line and provided with a straight line at right angles to the said one line calibrated in the same units, the base member being provided with a table of values of the lengths of the sides of squares equivalent to circles having diameters corresponding to the calibrations of the said one line, the said values being tabulated adjacent to the corresponding calibrations, and the base member being provided also with a table of values of the corresponding areas of the circles and the equivalent squares.

' 7. An instrument for approximately squaring a circle having, in combination, a rectangular base member having a front face and a rear face and having projecting end portions one of which is provided with a plane face perpendicular to the plane of the rectangle and parallel to its shorter sides, a rectangular indicator member disposed at right angles to the rectangular base member and having a projecting portion having a plane face parallel to the plane face of the said one projecting end portion, the indicator member being provided with a U-shaped recess for slidably receiving the base member, the base member being provided with two straight lines inclined to each other at an angle determined by the equation where 7r is the ratio of the circumference to the diameter of a circle, one of the lines being linearly calibrated in predetermined units, and the indicator member being mounted to slide in a plane parallel to the plane of the rectangular base member substantially parallel to the said one line and being provided with a straight line at right angles to the said one line calibrated in the same units.

WILLIAM BALVAIRD THOMSON.

' REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Number Name Date 1,200,569, Young Oct. 10, 1916 1,553,683 Furbish Sept. 15, 1925 2,300,448 Ludwig Nov. 3, 1942 

